Hi!
Strollers are available for rent at numerous places throughtout Walt Disney World. We did it for several years and it was pretty painless except sometimes when a park first opened; there could be a long line for stroller rentals when you would rather be on an attraction! Quantities are limited and subject to availability, but we were always able to rent one when we wanted to and we've been there during some very busy times!
The current prices can be found
here. It's not exactly inexpensive in my opinion, but Disney does offer a Length of Stay rental ticket that costs a little less if you were planning on renting each day. If you visit another location on the same day, you do not have to rent another stroller; for example, you could rent a stroller at the Magic Kingdom, leave it there, visit Epcot later that same day, and pick up a stroller there for free.
When we were not weighed down with too much luggage of our own, I always preferred taking our own stroller. Some other guests buy a stroller at their resort or buy one after getting to Florida but before getting to Walt Disney World; I've heard that they either ship them back home or just leave them behind for another lucky family!
You should also check with your airline about any fees. The fees, if any, might be different if you explain you will be pushing the stroller through security and to the front door of your plane, with your child in it, as opposed to checking it as checked luggage.
Good luck and enjoy your trip!